forked from AbsInt/CompCert
-
Notifications
You must be signed in to change notification settings - Fork 0
/
Copy pathSelectLongproof.v
555 lines (506 loc) · 23.8 KB
/
SelectLongproof.v
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70
71
72
73
74
75
76
77
78
79
80
81
82
83
84
85
86
87
88
89
90
91
92
93
94
95
96
97
98
99
100
101
102
103
104
105
106
107
108
109
110
111
112
113
114
115
116
117
118
119
120
121
122
123
124
125
126
127
128
129
130
131
132
133
134
135
136
137
138
139
140
141
142
143
144
145
146
147
148
149
150
151
152
153
154
155
156
157
158
159
160
161
162
163
164
165
166
167
168
169
170
171
172
173
174
175
176
177
178
179
180
181
182
183
184
185
186
187
188
189
190
191
192
193
194
195
196
197
198
199
200
201
202
203
204
205
206
207
208
209
210
211
212
213
214
215
216
217
218
219
220
221
222
223
224
225
226
227
228
229
230
231
232
233
234
235
236
237
238
239
240
241
242
243
244
245
246
247
248
249
250
251
252
253
254
255
256
257
258
259
260
261
262
263
264
265
266
267
268
269
270
271
272
273
274
275
276
277
278
279
280
281
282
283
284
285
286
287
288
289
290
291
292
293
294
295
296
297
298
299
300
301
302
303
304
305
306
307
308
309
310
311
312
313
314
315
316
317
318
319
320
321
322
323
324
325
326
327
328
329
330
331
332
333
334
335
336
337
338
339
340
341
342
343
344
345
346
347
348
349
350
351
352
353
354
355
356
357
358
359
360
361
362
363
364
365
366
367
368
369
370
371
372
373
374
375
376
377
378
379
380
381
382
383
384
385
386
387
388
389
390
391
392
393
394
395
396
397
398
399
400
401
402
403
404
405
406
407
408
409
410
411
412
413
414
415
416
417
418
419
420
421
422
423
424
425
426
427
428
429
430
431
432
433
434
435
436
437
438
439
440
441
442
443
444
445
446
447
448
449
450
451
452
453
454
455
456
457
458
459
460
461
462
463
464
465
466
467
468
469
470
471
472
473
474
475
476
477
478
479
480
481
482
483
484
485
486
487
488
489
490
491
492
493
494
495
496
497
498
499
500
501
502
503
504
505
506
507
508
509
510
511
512
513
514
515
516
517
518
519
520
521
522
523
524
525
526
527
528
529
530
531
532
533
534
535
536
537
538
539
540
541
542
543
544
545
546
547
548
549
550
551
552
553
554
555
(* *********************************************************************)
(* *)
(* The Compcert verified compiler *)
(* *)
(* Xavier Leroy, INRIA Paris *)
(* *)
(* Copyright Institut National de Recherche en Informatique et en *)
(* Automatique. All rights reserved. This file is distributed *)
(* under the terms of the INRIA Non-Commercial License Agreement. *)
(* *)
(* *********************************************************************)
(** Correctness of instruction selection for 64-bit integer operations *)
Require Import String Coqlib Maps Integers Floats Errors.
Require Archi.
Require Import AST Values Memory Globalenvs Events.
Require Import Cminor Op CminorSel.
Require Import SelectOp SelectOpproof SplitLong SplitLongproof.
Require Import SelectLong.
Local Open Scope cminorsel_scope.
Local Open Scope string_scope.
(** * Correctness of the instruction selection functions for 64-bit operators *)
Section CMCONSTR.
Variable prog: program.
Variable hf: helper_functions.
Hypothesis HELPERS: helper_functions_declared prog hf.
Let ge := Genv.globalenv prog.
Variable sp: val.
Variable e: env.
Variable m: mem.
Definition unary_constructor_sound (cstr: expr -> expr) (sem: val -> val) : Prop :=
forall le a x,
eval_expr ge sp e m le a x ->
exists v, eval_expr ge sp e m le (cstr a) v /\ Val.lessdef (sem x) v.
Definition binary_constructor_sound (cstr: expr -> expr -> expr) (sem: val -> val -> val) : Prop :=
forall le a x b y,
eval_expr ge sp e m le a x ->
eval_expr ge sp e m le b y ->
exists v, eval_expr ge sp e m le (cstr a b) v /\ Val.lessdef (sem x y) v.
Definition partial_unary_constructor_sound (cstr: expr -> expr) (sem: val -> option val) : Prop :=
forall le a x y,
eval_expr ge sp e m le a x ->
sem x = Some y ->
exists v, eval_expr ge sp e m le (cstr a) v /\ Val.lessdef y v.
Definition partial_binary_constructor_sound (cstr: expr -> expr -> expr) (sem: val -> val -> option val) : Prop :=
forall le a x b y z,
eval_expr ge sp e m le a x ->
eval_expr ge sp e m le b y ->
sem x y = Some z ->
exists v, eval_expr ge sp e m le (cstr a b) v /\ Val.lessdef z v.
Theorem eval_longconst:
forall le n, eval_expr ge sp e m le (longconst n) (Vlong n).
Proof.
unfold longconst; intros; destruct Archi.splitlong.
apply SplitLongproof.eval_longconst.
EvalOp.
Qed.
Lemma is_longconst_sound:
forall v a n le,
is_longconst a = Some n -> eval_expr ge sp e m le a v -> v = Vlong n.
Proof with (try discriminate).
intros. unfold is_longconst in *. destruct Archi.splitlong.
eapply SplitLongproof.is_longconst_sound; eauto.
assert (a = Eop (Olongconst n) Enil).
{ destruct a... destruct o... destruct e0... congruence. }
subst a. InvEval. auto.
Qed.
Theorem eval_intoflong: unary_constructor_sound intoflong Val.loword.
Proof.
unfold intoflong; destruct Archi.splitlong. apply SplitLongproof.eval_intoflong.
red; intros. destruct (is_longconst a) as [n|] eqn:C.
- TrivialExists. simpl. erewrite (is_longconst_sound x) by eauto. auto.
- TrivialExists.
Qed.
Theorem eval_longofintu: unary_constructor_sound longofintu Val.longofintu.
Proof.
unfold longofintu; destruct Archi.splitlong. apply SplitLongproof.eval_longofintu.
red; intros. destruct (is_intconst a) as [n|] eqn:C.
- econstructor; split. apply eval_longconst.
exploit is_intconst_sound; eauto. intros; subst x. auto.
- TrivialExists.
Qed.
Theorem eval_longofint: unary_constructor_sound longofint Val.longofint.
Proof.
unfold longofint; destruct Archi.splitlong. apply SplitLongproof.eval_longofint.
red; intros. destruct (is_intconst a) as [n|] eqn:C.
- econstructor; split. apply eval_longconst.
exploit is_intconst_sound; eauto. intros; subst x. auto.
- TrivialExists.
Qed.
Theorem eval_notl: unary_constructor_sound notl Val.notl.
Proof.
unfold notl; destruct Archi.splitlong. apply SplitLongproof.eval_notl.
red; intros. destruct (notl_match a).
- InvEval. econstructor; split. apply eval_longconst. auto.
- InvEval. subst. exists v1; split; auto. destruct v1; simpl; auto. rewrite Int64.not_involutive; auto.
- TrivialExists.
Qed.
Theorem eval_andlimm: forall n, unary_constructor_sound (andlimm n) (fun v => Val.andl v (Vlong n)).
Proof.
unfold andlimm; intros; red; intros.
predSpec Int64.eq Int64.eq_spec n Int64.zero.
exists (Vlong Int64.zero); split. apply eval_longconst.
subst. destruct x; simpl; auto. rewrite Int64.and_zero; auto.
predSpec Int64.eq Int64.eq_spec n Int64.mone.
exists x; split. assumption.
subst. destruct x; simpl; auto. rewrite Int64.and_mone; auto.
destruct (andlimm_match a); InvEval; subst.
- econstructor; split. apply eval_longconst. simpl. rewrite Int64.and_commut; auto.
- TrivialExists. simpl. rewrite Val.andl_assoc. rewrite Int64.and_commut; auto.
- TrivialExists.
Qed.
Theorem eval_andl: binary_constructor_sound andl Val.andl.
Proof.
unfold andl; destruct Archi.splitlong. apply SplitLongproof.eval_andl.
red; intros. destruct (andl_match a b).
- InvEval. rewrite Val.andl_commut. apply eval_andlimm; auto.
- InvEval. apply eval_andlimm; auto.
- TrivialExists.
Qed.
Theorem eval_orlimm: forall n, unary_constructor_sound (orlimm n) (fun v => Val.orl v (Vlong n)).
Proof.
unfold orlimm; intros; red; intros.
predSpec Int64.eq Int64.eq_spec n Int64.zero.
exists x; split; auto. subst. destruct x; simpl; auto. rewrite Int64.or_zero; auto.
predSpec Int64.eq Int64.eq_spec n Int64.mone.
econstructor; split. apply eval_longconst. subst. destruct x; simpl; auto. rewrite Int64.or_mone; auto.
destruct (orlimm_match a); InvEval; subst.
- econstructor; split. apply eval_longconst. simpl. rewrite Int64.or_commut; auto.
- TrivialExists. simpl. rewrite Val.orl_assoc. rewrite Int64.or_commut; auto.
- TrivialExists.
Qed.
Theorem eval_orl: binary_constructor_sound orl Val.orl.
Proof.
unfold orl; destruct Archi.splitlong. apply SplitLongproof.eval_orl.
red; intros.
assert (DEFAULT: exists v, eval_expr ge sp e m le (Eop Oorl (a:::b:::Enil)) v /\ Val.lessdef (Val.orl x y) v) by TrivialExists.
assert (ROR: forall v n1 n2,
Int.add n1 n2 = Int64.iwordsize' ->
Val.lessdef (Val.orl (Val.shll v (Vint n1)) (Val.shrlu v (Vint n2)))
(Val.rorl v (Vint n2))).
{ intros. destruct v; simpl; auto.
destruct (Int.ltu n1 Int64.iwordsize') eqn:N1; auto.
destruct (Int.ltu n2 Int64.iwordsize') eqn:N2; auto.
simpl. rewrite <- Int64.or_ror'; auto. }
destruct (orl_match a b).
- InvEval. rewrite Val.orl_commut. apply eval_orlimm; auto.
- InvEval. apply eval_orlimm; auto.
- predSpec Int.eq Int.eq_spec (Int.add n1 n2) Int64.iwordsize'; auto.
destruct (same_expr_pure t1 t2) eqn:?; auto.
InvEval. exploit eval_same_expr; eauto. intros [EQ1 EQ2]; subst.
exists (Val.rorl v0 (Vint n2)); split. EvalOp. apply ROR; auto.
- predSpec Int.eq Int.eq_spec (Int.add n1 n2) Int64.iwordsize'; auto.
destruct (same_expr_pure t1 t2) eqn:?; auto.
InvEval. exploit eval_same_expr; eauto. intros [EQ1 EQ2]; subst.
exists (Val.rorl v1 (Vint n2)); split. EvalOp. rewrite Val.orl_commut. apply ROR; auto.
- apply DEFAULT.
Qed.
Theorem eval_xorlimm: forall n, unary_constructor_sound (xorlimm n) (fun v => Val.xorl v (Vlong n)).
Proof.
unfold xorlimm; intros; red; intros.
predSpec Int64.eq Int64.eq_spec n Int64.zero.
exists x; split; auto. subst. destruct x; simpl; auto. rewrite Int64.xor_zero; auto.
predSpec Int64.eq Int64.eq_spec n Int64.mone.
replace (Val.xorl x (Vlong n)) with (Val.notl x). apply eval_notl; auto.
subst n. destruct x; simpl; auto.
destruct (xorlimm_match a); InvEval; subst.
- econstructor; split. apply eval_longconst. simpl. rewrite Int64.xor_commut; auto.
- TrivialExists. simpl. rewrite Val.xorl_assoc. rewrite Int64.xor_commut; auto.
- TrivialExists. simpl. destruct v1; simpl; auto. unfold Int64.not.
rewrite Int64.xor_assoc. apply f_equal. apply f_equal. apply f_equal.
apply Int64.xor_commut.
- TrivialExists.
Qed.
Theorem eval_xorl: binary_constructor_sound xorl Val.xorl.
Proof.
unfold xorl; destruct Archi.splitlong. apply SplitLongproof.eval_xorl.
red; intros. destruct (xorl_match a b).
- InvEval. rewrite Val.xorl_commut. apply eval_xorlimm; auto.
- InvEval. apply eval_xorlimm; auto.
- TrivialExists.
Qed.
Theorem eval_shllimm: forall n, unary_constructor_sound (fun e => shllimm e n) (fun v => Val.shll v (Vint n)).
Proof.
intros; unfold shllimm. destruct Archi.splitlong eqn:SL. apply SplitLongproof.eval_shllimm; auto.
red; intros.
predSpec Int.eq Int.eq_spec n Int.zero.
exists x; split; auto. subst n; destruct x; simpl; auto.
destruct (Int.ltu Int.zero Int64.iwordsize'); auto.
change (Int64.shl' i Int.zero) with (Int64.shl i Int64.zero). rewrite Int64.shl_zero; auto.
destruct (Int.ltu n Int64.iwordsize') eqn:LT; simpl.
assert (DEFAULT: exists v, eval_expr ge sp e m le (Eop (Oshllimm n) (a:::Enil)) v
/\ Val.lessdef (Val.shll x (Vint n)) v) by TrivialExists.
destruct (shllimm_match a); InvEval.
- TrivialExists. simpl; rewrite LT; auto.
- destruct (Int.ltu (Int.add n n1) Int64.iwordsize') eqn:LT'; auto.
subst. econstructor; split. EvalOp. simpl; eauto.
destruct v1; simpl; auto. rewrite LT'.
destruct (Int.ltu n1 Int64.iwordsize') eqn:LT1; auto.
simpl; rewrite LT. rewrite Int.add_commut, Int64.shl'_shl'; auto. rewrite Int.add_commut; auto.
- destruct (shift_is_scale n); auto.
TrivialExists. simpl. destruct v1; simpl; auto.
rewrite LT. rewrite ! Int64.repr_unsigned. rewrite Int64.shl'_one_two_p.
rewrite ! Int64.shl'_mul_two_p. rewrite Int64.mul_add_distr_l. auto.
- destruct (shift_is_scale n); auto.
TrivialExists. simpl. destruct x; simpl; auto.
rewrite LT. rewrite ! Int64.repr_unsigned. rewrite Int64.shl'_one_two_p.
rewrite ! Int64.shl'_mul_two_p. rewrite Int64.add_zero. auto.
- TrivialExists. constructor; eauto. constructor. EvalOp. simpl; eauto. constructor. auto.
Qed.
Theorem eval_shrluimm: forall n, unary_constructor_sound (fun e => shrluimm e n) (fun v => Val.shrlu v (Vint n)).
Proof.
intros; unfold shrluimm. destruct Archi.splitlong eqn:SL. apply SplitLongproof.eval_shrluimm; auto.
red; intros.
predSpec Int.eq Int.eq_spec n Int.zero.
exists x; split; auto. subst n; destruct x; simpl; auto.
destruct (Int.ltu Int.zero Int64.iwordsize'); auto.
change (Int64.shru' i Int.zero) with (Int64.shru i Int64.zero). rewrite Int64.shru_zero; auto.
destruct (Int.ltu n Int64.iwordsize') eqn:LT; simpl.
assert (DEFAULT: exists v, eval_expr ge sp e m le (Eop (Oshrluimm n) (a:::Enil)) v
/\ Val.lessdef (Val.shrlu x (Vint n)) v) by TrivialExists.
destruct (shrluimm_match a); InvEval.
- TrivialExists. simpl; rewrite LT; auto.
- destruct (Int.ltu (Int.add n n1) Int64.iwordsize') eqn:LT'; auto.
subst. econstructor; split. EvalOp. simpl; eauto.
destruct v1; simpl; auto. rewrite LT'.
destruct (Int.ltu n1 Int64.iwordsize') eqn:LT1; auto.
simpl; rewrite LT. rewrite Int.add_commut, Int64.shru'_shru'; auto. rewrite Int.add_commut; auto.
- apply DEFAULT.
- TrivialExists. constructor; eauto. constructor. EvalOp. simpl; eauto. constructor. auto.
Qed.
Theorem eval_shrlimm: forall n, unary_constructor_sound (fun e => shrlimm e n) (fun v => Val.shrl v (Vint n)).
Proof.
intros; unfold shrlimm. destruct Archi.splitlong eqn:SL. apply SplitLongproof.eval_shrlimm; auto.
red; intros.
predSpec Int.eq Int.eq_spec n Int.zero.
exists x; split; auto. subst n; destruct x; simpl; auto.
destruct (Int.ltu Int.zero Int64.iwordsize'); auto.
change (Int64.shr' i Int.zero) with (Int64.shr i Int64.zero). rewrite Int64.shr_zero; auto.
destruct (Int.ltu n Int64.iwordsize') eqn:LT; simpl.
assert (DEFAULT: exists v, eval_expr ge sp e m le (Eop (Oshrlimm n) (a:::Enil)) v
/\ Val.lessdef (Val.shrl x (Vint n)) v) by TrivialExists.
destruct (shrlimm_match a); InvEval.
- TrivialExists. simpl; rewrite LT; auto.
- destruct (Int.ltu (Int.add n n1) Int64.iwordsize') eqn:LT'; auto.
subst. econstructor; split. EvalOp. simpl; eauto.
destruct v1; simpl; auto. rewrite LT'.
destruct (Int.ltu n1 Int64.iwordsize') eqn:LT1; auto.
simpl; rewrite LT. rewrite Int.add_commut, Int64.shr'_shr'; auto. rewrite Int.add_commut; auto.
- apply DEFAULT.
- TrivialExists. constructor; eauto. constructor. EvalOp. simpl; eauto. constructor. auto.
Qed.
Theorem eval_shll: binary_constructor_sound shll Val.shll.
Proof.
unfold shll. destruct Archi.splitlong eqn:SL. apply SplitLongproof.eval_shll; auto.
red; intros. destruct (is_intconst b) as [n2|] eqn:C.
- exploit is_intconst_sound; eauto. intros EQ; subst y. apply eval_shllimm; auto.
- TrivialExists.
Qed.
Theorem eval_shrlu: binary_constructor_sound shrlu Val.shrlu.
Proof.
unfold shrlu. destruct Archi.splitlong eqn:SL. apply SplitLongproof.eval_shrlu; auto.
red; intros. destruct (is_intconst b) as [n2|] eqn:C.
- exploit is_intconst_sound; eauto. intros EQ; subst y. apply eval_shrluimm; auto.
- TrivialExists.
Qed.
Theorem eval_shrl: binary_constructor_sound shrl Val.shrl.
Proof.
unfold shrl. destruct Archi.splitlong eqn:SL. apply SplitLongproof.eval_shrl; auto.
red; intros. destruct (is_intconst b) as [n2|] eqn:C.
- exploit is_intconst_sound; eauto. intros EQ; subst y. apply eval_shrlimm; auto.
- TrivialExists.
Qed.
Theorem eval_negl: unary_constructor_sound negl Val.negl.
Proof.
unfold negl. destruct Archi.splitlong eqn:SL. apply SplitLongproof.eval_negl; auto.
red; intros. destruct (is_longconst a) as [n|] eqn:C.
- exploit is_longconst_sound; eauto. intros EQ; subst x.
econstructor; split. apply eval_longconst. auto.
- TrivialExists.
Qed.
Theorem eval_addlimm: forall n, unary_constructor_sound (addlimm n) (fun v => Val.addl v (Vlong n)).
Proof.
unfold addlimm; intros; red; intros.
predSpec Int64.eq Int64.eq_spec n Int64.zero.
subst. exists x; split; auto.
destruct x; simpl; rewrite ?Int64.add_zero, ?Ptrofs.add_zero; auto.
destruct (addlimm_match a); InvEval.
- econstructor; split. apply eval_longconst. rewrite Int64.add_commut; auto.
- inv H. simpl in H6. TrivialExists. simpl.
erewrite eval_offset_addressing_total_64 by eauto. rewrite Int64.repr_signed; auto.
- TrivialExists. simpl. rewrite Int64.repr_signed; auto.
Qed.
Theorem eval_addl: binary_constructor_sound addl Val.addl.
Proof.
assert (A: forall x y, Int64.repr (x + y) = Int64.add (Int64.repr x) (Int64.repr y)).
{ intros; apply Int64.eqm_samerepr; auto with ints. }
assert (B: forall id ofs n, Archi.ptr64 = true ->
Genv.symbol_address ge id (Ptrofs.add ofs (Ptrofs.repr n)) =
Val.addl (Genv.symbol_address ge id ofs) (Vlong (Int64.repr n))).
{ intros. replace (Ptrofs.repr n) with (Ptrofs.of_int64 (Int64.repr n)) by auto with ptrofs.
apply Genv.shift_symbol_address_64; auto. }
unfold addl. destruct Archi.splitlong eqn:SL.
apply SplitLongproof.eval_addl. apply Archi.splitlong_ptr32; auto.
red; intros; destruct (addl_match a b); InvEval.
- rewrite Val.addl_commut. apply eval_addlimm; auto.
- apply eval_addlimm; auto.
- subst. TrivialExists. simpl. rewrite A, Val.addl_permut_4. auto.
- subst. TrivialExists. simpl. rewrite A, Val.addl_assoc. decEq; decEq. rewrite Val.addl_permut. auto.
- subst. TrivialExists. simpl. rewrite A, Val.addl_permut_4. rewrite <- Val.addl_permut. rewrite <- Val.addl_assoc. auto.
- subst. TrivialExists. simpl. rewrite Val.addl_commut; auto.
- subst. TrivialExists.
- subst. TrivialExists. simpl. rewrite ! Val.addl_assoc. rewrite (Val.addl_commut y). auto.
- subst. TrivialExists. simpl. rewrite ! Val.addl_assoc. auto.
- TrivialExists. simpl.
unfold Val.addl. destruct Archi.ptr64, x, y; auto.
+ rewrite Int64.add_zero; auto.
+ rewrite Ptrofs.add_assoc, Ptrofs.add_zero. auto.
+ rewrite Ptrofs.add_assoc, Ptrofs.add_zero. auto.
+ rewrite Int64.add_zero; auto.
Qed.
Theorem eval_subl: binary_constructor_sound subl Val.subl.
Proof.
unfold subl. destruct Archi.splitlong eqn:SL.
apply SplitLongproof.eval_subl. apply Archi.splitlong_ptr32; auto.
red; intros; destruct (subl_match a b); InvEval.
- rewrite Val.subl_addl_opp. apply eval_addlimm; auto.
- subst. rewrite Val.subl_addl_l. rewrite Val.subl_addl_r.
rewrite Val.addl_assoc. simpl. rewrite Int64.add_commut. rewrite <- Int64.sub_add_opp.
replace (Int64.repr (n1 - n2)) with (Int64.sub (Int64.repr n1) (Int64.repr n2)).
apply eval_addlimm; EvalOp.
apply Int64.eqm_samerepr; auto with ints.
- subst. rewrite Val.subl_addl_l. apply eval_addlimm; EvalOp.
- subst. rewrite Val.subl_addl_r.
replace (Int64.repr (-n2)) with (Int64.neg (Int64.repr n2)).
apply eval_addlimm; EvalOp.
apply Int64.eqm_samerepr; auto with ints.
- TrivialExists.
Qed.
Theorem eval_mullimm_base: forall n, unary_constructor_sound (mullimm_base n) (fun v => Val.mull v (Vlong n)).
Proof.
intros; unfold mullimm_base. red; intros.
generalize (Int64.one_bits'_decomp n); intros D.
destruct (Int64.one_bits' n) as [ | i [ | j [ | ? ? ]]] eqn:B.
- TrivialExists.
- replace (Val.mull x (Vlong n)) with (Val.shll x (Vint i)).
apply eval_shllimm; auto.
simpl in D. rewrite D, Int64.add_zero. destruct x; simpl; auto.
rewrite (Int64.one_bits'_range n) by (rewrite B; auto with coqlib).
rewrite Int64.shl'_mul; auto.
- set (le' := x :: le).
assert (A0: eval_expr ge sp e m le' (Eletvar O) x) by (constructor; reflexivity).
exploit (eval_shllimm i). eexact A0. intros (v1 & A1 & B1).
exploit (eval_shllimm j). eexact A0. intros (v2 & A2 & B2).
exploit (eval_addl). eexact A1. eexact A2. intros (v3 & A3 & B3).
exists v3; split. econstructor; eauto.
rewrite D. simpl. rewrite Int64.add_zero. destruct x; auto.
simpl in *.
rewrite (Int64.one_bits'_range n) in B1 by (rewrite B; auto with coqlib).
rewrite (Int64.one_bits'_range n) in B2 by (rewrite B; auto with coqlib).
inv B1; inv B2. simpl in B3; inv B3.
rewrite Int64.mul_add_distr_r. rewrite <- ! Int64.shl'_mul. auto.
- TrivialExists.
Qed.
Theorem eval_mullimm: forall n, unary_constructor_sound (mullimm n) (fun v => Val.mull v (Vlong n)).
Proof.
unfold mullimm. intros; red; intros.
destruct Archi.splitlong eqn:SL.
eapply SplitLongproof.eval_mullimm; eauto.
predSpec Int64.eq Int64.eq_spec n Int64.zero.
exists (Vlong Int64.zero); split. apply eval_longconst.
destruct x; simpl; auto. subst n; rewrite Int64.mul_zero; auto.
predSpec Int64.eq Int64.eq_spec n Int64.one.
exists x; split; auto.
destruct x; simpl; auto. subst n; rewrite Int64.mul_one; auto.
destruct (mullimm_match a); InvEval.
- econstructor; split. apply eval_longconst. rewrite Int64.mul_commut; auto.
- exploit (eval_mullimm_base n); eauto. intros (v2 & A2 & B2).
exploit (eval_addlimm (Int64.mul n (Int64.repr n2))). eexact A2. intros (v3 & A3 & B3).
exists v3; split; auto.
destruct v1; simpl; auto.
simpl in B2; inv B2. simpl in B3; inv B3. rewrite Int64.mul_add_distr_l.
rewrite (Int64.mul_commut n). auto.
- apply eval_mullimm_base; auto.
Qed.
Theorem eval_mull: binary_constructor_sound mull Val.mull.
Proof.
unfold mull. destruct Archi.splitlong eqn:SL. apply SplitLongproof.eval_mull; auto.
red; intros; destruct (mull_match a b); InvEval.
- rewrite Val.mull_commut. apply eval_mullimm; auto.
- apply eval_mullimm; auto.
- TrivialExists.
Qed.
Theorem eval_mullhu:
forall n, unary_constructor_sound (fun a => mullhu a n) (fun v => Val.mullhu v (Vlong n)).
Proof.
unfold mullhu; intros. destruct Archi.splitlong eqn:SL. apply SplitLongproof.eval_mullhu; auto.
red; intros. TrivialExists. constructor. eauto. constructor. apply eval_longconst. constructor. auto.
Qed.
Theorem eval_mullhs:
forall n, unary_constructor_sound (fun a => mullhs a n) (fun v => Val.mullhs v (Vlong n)).
Proof.
unfold mullhs; intros. destruct Archi.splitlong eqn:SL. apply SplitLongproof.eval_mullhs; auto.
red; intros. TrivialExists. constructor. eauto. constructor. apply eval_longconst. constructor. auto.
Qed.
Theorem eval_shrxlimm:
forall le a n x z,
eval_expr ge sp e m le a x ->
Val.shrxl x (Vint n) = Some z ->
exists v, eval_expr ge sp e m le (shrxlimm a n) v /\ Val.lessdef z v.
Proof.
unfold shrxlimm; intros. destruct Archi.splitlong eqn:SL.
+ eapply SplitLongproof.eval_shrxlimm; eauto using Archi.splitlong_ptr32.
+ predSpec Int.eq Int.eq_spec n Int.zero.
- subst n. destruct x; simpl in H0; inv H0. econstructor; split; eauto.
change (Int.ltu Int.zero (Int.repr 63)) with true. simpl. rewrite Int64.shrx'_zero; auto.
- TrivialExists.
Qed.
Theorem eval_divls_base: partial_binary_constructor_sound divls_base Val.divls.
Proof.
unfold divls_base; red; intros. destruct Archi.splitlong eqn:SL.
eapply SplitLongproof.eval_divls_base; eauto.
TrivialExists.
Qed.
Theorem eval_modls_base: partial_binary_constructor_sound modls_base Val.modls.
Proof.
unfold modls_base; red; intros. destruct Archi.splitlong eqn:SL.
eapply SplitLongproof.eval_modls_base; eauto.
TrivialExists.
Qed.
Theorem eval_divlu_base: partial_binary_constructor_sound divlu_base Val.divlu.
Proof.
unfold divlu_base; red; intros. destruct Archi.splitlong eqn:SL.
eapply SplitLongproof.eval_divlu_base; eauto.
TrivialExists.
Qed.
Theorem eval_modlu_base: partial_binary_constructor_sound modlu_base Val.modlu.
Proof.
unfold modlu_base; red; intros. destruct Archi.splitlong eqn:SL.
eapply SplitLongproof.eval_modlu_base; eauto.
TrivialExists.
Qed.
Theorem eval_cmplu:
forall c le a x b y v,
eval_expr ge sp e m le a x ->
eval_expr ge sp e m le b y ->
Val.cmplu (Mem.valid_pointer m) c x y = Some v ->
eval_expr ge sp e m le (cmplu c a b) v.
Proof.
unfold cmplu; intros. destruct Archi.splitlong eqn:SL.
eapply SplitLongproof.eval_cmplu; eauto using Archi.splitlong_ptr32.
unfold Val.cmplu in H1.
destruct (Val.cmplu_bool (Mem.valid_pointer m) c x y) as [vb|] eqn:C; simpl in H1; inv H1.
destruct (is_longconst a) as [n1|] eqn:LC1; destruct (is_longconst b) as [n2|] eqn:LC2;
try (assert (x = Vlong n1) by (eapply is_longconst_sound; eauto));
try (assert (y = Vlong n2) by (eapply is_longconst_sound; eauto));
subst.
- simpl in C; inv C. EvalOp. destruct (Int64.cmpu c n1 n2); reflexivity.
- EvalOp. simpl. rewrite Val.swap_cmplu_bool. rewrite C; auto.
- EvalOp. simpl; rewrite C; auto.
- EvalOp. simpl; rewrite C; auto.
Qed.
Theorem eval_cmpl:
forall c le a x b y v,
eval_expr ge sp e m le a x ->
eval_expr ge sp e m le b y ->
Val.cmpl c x y = Some v ->
eval_expr ge sp e m le (cmpl c a b) v.
Proof.
unfold cmpl; intros. destruct Archi.splitlong eqn:SL.
eapply SplitLongproof.eval_cmpl; eauto.
unfold Val.cmpl in H1.
destruct (Val.cmpl_bool c x y) as [vb|] eqn:C; simpl in H1; inv H1.
destruct (is_longconst a) as [n1|] eqn:LC1; destruct (is_longconst b) as [n2|] eqn:LC2;
try (assert (x = Vlong n1) by (eapply is_longconst_sound; eauto));
try (assert (y = Vlong n2) by (eapply is_longconst_sound; eauto));
subst.
- simpl in C; inv C. EvalOp. destruct (Int64.cmp c n1 n2); reflexivity.
- EvalOp. simpl. rewrite Val.swap_cmpl_bool. rewrite C; auto.
- EvalOp. simpl; rewrite C; auto.
- EvalOp. simpl; rewrite C; auto.
Qed.
Theorem eval_longoffloat: partial_unary_constructor_sound longoffloat Val.longoffloat.
Proof.
unfold longoffloat; red; intros. destruct Archi.splitlong eqn:SL.
eapply SplitLongproof.eval_longoffloat; eauto.
TrivialExists.
Qed.
Theorem eval_floatoflong: partial_unary_constructor_sound floatoflong Val.floatoflong.
Proof.
unfold floatoflong; red; intros. destruct Archi.splitlong eqn:SL.
eapply SplitLongproof.eval_floatoflong; eauto.
TrivialExists.
Qed.
Theorem eval_longofsingle: partial_unary_constructor_sound longofsingle Val.longofsingle.
Proof.
unfold longofsingle; red; intros. destruct Archi.splitlong eqn:SL.
eapply SplitLongproof.eval_longofsingle; eauto.
TrivialExists.
Qed.
Theorem eval_singleoflong: partial_unary_constructor_sound singleoflong Val.singleoflong.
Proof.
unfold singleoflong; red; intros. destruct Archi.splitlong eqn:SL.
eapply SplitLongproof.eval_singleoflong; eauto.
TrivialExists.
Qed.
End CMCONSTR.